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- The '''Hartree equation''' is of the form ...c Schrodinger equation; the [[cubic NLS]] is in some sense a limit of this equation as <math>n \rightarrow d,</math> (after suitable normalization of the kerne3 KB (469 words) - 17:30, 9 July 2007
- #REDIRECT [[Semilinear Schrodinger equation]]45 bytes (4 words) - 03:05, 29 July 2006
- 29 bytes (3 words) - 06:34, 27 July 2006
- 32 bytes (3 words) - 00:45, 10 August 2006
- 31 bytes (3 words) - 07:09, 31 July 2006
- The (homogeneous) '''Airy equation''' is given by ...tions. One such problem is the reduced (or linearized) [[Korteweg-de Vries equation]]1 KB (214 words) - 22:11, 19 January 2018
- The theory of the '''Schrodinger equation with potential'''285 bytes (43 words) - 04:51, 4 August 2006
- ...equation]], and when <math>\lambda > 0</math> the equation is the [[KP-II equation]]. These equations are a model for shallow long waves in the x direction, ...varying amplitude, weak non-linearity) one can obtain the Davey-Stewatson equation (formally, at least). KP-I also arises as a model for sound waves in ferro3 KB (527 words) - 19:46, 3 February 2011
- 23 bytes (3 words) - 06:38, 29 July 2006
- 34 bytes (3 words) - 15:45, 5 May 2007
- The '''free Schrodinger equation''' ...dimensions. One can also consider the inhomogeneous (forced) Schrodinger equation2 KB (307 words) - 07:45, 15 June 2009
- 38 bytes (4 words) - 17:01, 30 July 2006
- The '''free wave equation''' on <math>{\mathbb R}^{1+d}</math> is given by One can add a mass term to create the [[Klein-Gordon equation]].2 KB (358 words) - 16:00, 1 July 2018
- The '''Klein-Gordon equation''' is given by [[Minkowski space]]. When the mass is zero this is just the [[free wave equation]]. When the mass is positive, one usually rescales spacetime to normalize1 KB (169 words) - 20:56, 10 June 2007
- 29 bytes (3 words) - 22:12, 3 September 2007
- 29 bytes (3 words) - 22:12, 3 September 2007
- The (hyperbolic) '''minimal surface equation''' takes the form ...n \times R</math> ). This is the Minkowski analogue of the minimal surface equation in Euclidean space, see [[Hp1994]].809 bytes (125 words) - 22:10, 10 April 2007
- ...ints of that functional. It can be computed formally by starting with the equation ...iltonian]] can (in principle, at least) be expressed as the Euler-Lagrange equation of a functional, and conversely Euler-Lagrange equations can in principle b657 bytes (95 words) - 20:14, 11 June 2007
- The '''sine-Gordon equation''' ...ontinuum of pendula hanging from a wire. It is a [[completely integrable]] equation, and has many interesting solutions, including "breather" solutions.705 bytes (103 words) - 23:39, 22 January 2009
- ...nt the electromagnetic field. x and v live in R^3 and t lives in R. This equation is a coupled wave and conservation law system, and models collision-less pl ...\rho</math>. Considerably more is known for the existence theory of this equation.2 KB (297 words) - 13:33, 11 July 2007
- The '''KP-I equation''' is the special case of the [[Kadomtsev-Petviashvili equation]] when the parameter <math>\lambda</math> is negative. Well-posedness is u *If one considers the fifth-order KP-I equation (replace <math>u_{xxx}</math> by <math>u_{xxxxx}</math>) then one has GWP i3 KB (537 words) - 06:21, 21 July 2007
- #REDIRECT [[Kadomtsev-Petviashvili equation]]45 bytes (3 words) - 04:59, 27 July 2006
- The '''KP-II equation''' is the special case of the [[Kadomtsev-Petviashvili equation]] when the The KP-II equation can be generalized to three dimensions (replace partial_yy with partial_yy2 KB (264 words) - 01:22, 17 March 2007
- <center>'''Benjamin-Ono equation'''</center> The Benjamin-Ono equation (BO) [[Bj1967]], [[On1975]], which models one-dimensional internal waves in4 KB (707 words) - 18:49, 25 October 2008
- {{equation | equation = <math>u_t + u_{xxx} = \pm u^3 u_x</math>2 KB (376 words) - 22:26, 4 March 2007
- {{equation | equation = <math>u_t + u_{xxx} = \pm u^4 u_x</math>3 KB (496 words) - 22:03, 4 March 2007
- {{equation | equation = <math>iu_t + \Delta u = \pm |u|^{p-1} u</math>2 KB (349 words) - 05:44, 21 July 2007
- #REDIRECT [[Hartree equation]]30 bytes (3 words) - 04:29, 29 July 2006
- 29 bytes (3 words) - 22:12, 3 September 2007
- The '''Korteweg-de Vries (KdV) equation''' is The equation is [[completely integrable]], and has infinitely many conserved quantities.2 KB (307 words) - 16:10, 14 September 2006
- #REDIRECT [[Ginzburg-Landau-Schrodinger equation]]50 bytes (3 words) - 06:36, 16 January 2007
- 44 bytes (3 words) - 15:47, 5 May 2007
- 34 bytes (5 words) - 06:35, 27 July 2006
- The '''Ginzburg-Landau-Schrodinger''' equation is The main focus of study for this equation is the formation of vortices and their dynamics in the limit <math>\epsilon2 KB (368 words) - 07:45, 2 October 2007
- #REDIRECT [[Ginsberg-Landau-Schrodinger equation]]50 bytes (3 words) - 14:24, 9 August 2006
- 34 bytes (5 words) - 06:34, 27 July 2006
- The (defocusing) '''modified Korteweg-de Vries (mKdV) equation''' is ...roughly equivalent to the <span class="SpellE">H^k</span> norm of u. This equation has been studied [[modified Korteweg-de Vries on R|on the line]], [[modifie987 bytes (149 words) - 07:41, 31 July 2006
- ...particularly important class of such equations is the [[Schrodinger maps]] equation. ...ates which can recover a full derivative for the inhomogeneous Schrodinger equation providing there is sufficient decay in space, and so one can still obtain w5 KB (844 words) - 10:50, 24 August 2006
- 56 bytes (5 words) - 23:09, 14 August 2006
- 36 bytes (5 words) - 06:35, 27 July 2006
- 36 bytes (5 words) - 06:35, 27 July 2006
- 5 KB (777 words) - 20:49, 10 June 2007
- The local and global [[well-posedness]] theory for the [[Korteweg-de Vries equation]] on the real line is as follows. The <span class="SpellE">KdV</span> equation can also be generalized to a 2x2 system4 KB (617 words) - 21:51, 11 September 2007
- The local and global [[well-posedness]] theory for the [[Korteweg-de Vries equation]] on the torus is as follows. ...(at least in principle) from work on the <span class="SpellE">mKdV</span> equation by [Takaoka and Tsutsumi?]2 KB (386 words) - 22:51, 14 August 2006
- The local and global [[well-posedness]] theory for the [[Korteweg-de Vries equation]] on the half-line is as follows.2 KB (369 words) - 14:30, 10 August 2006
- '''Liouville's equation''' It is a limiting case of the [[Sine-Gordon|sinh-gordon equation]].2 KB (294 words) - 10:16, 27 January 2009
Page text matches
- A '''fully nonlinear''' equation takes the form for some nonlinear function ''F''. By differentiating this equation once and working with the system <math>v := (u, Du)</math>386 bytes (62 words) - 00:46, 8 August 2006
- * The [[Ginsberg-Landau-Schrodinger equation]] * The [[Benjamin-Ono equation]]832 bytes (85 words) - 14:27, 9 August 2006
- ...e vanishing mass limit of the [[Klein-Gordon equation]] is the [[free wave equation]].261 bytes (40 words) - 04:52, 4 August 2006
- ...proximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation262 bytes (28 words) - 20:59, 17 June 2009
- ...valence of a nonlinear Schrödinger equation and a Heisenberg ferromagnetic equation267 bytes (32 words) - 00:13, 11 July 2009
- ...dified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation297 bytes (36 words) - 00:10, 24 May 2009
- ...near estimates related with the KdV equation and the nonlinear Schrödinger equation317 bytes (37 words) - 18:01, 10 June 2009
- The '''Klein-Gordon equation''' is given by [[Minkowski space]]. When the mass is zero this is just the [[free wave equation]]. When the mass is positive, one usually rescales spacetime to normalize1 KB (169 words) - 20:56, 10 June 2007
- ...tion on the non-periodic domain <math>\mathbf{R}^d</math>, though when the equation has [[infinite speed of propagation]] then the justification of this limit430 bytes (70 words) - 04:56, 4 August 2006
- ...ely integrable in the sense that there exist a Lax pair formulation of the equation. a completely integrable equation. Some nonlinear PDE's as the KdV posed1 KB (166 words) - 08:40, 4 February 2011
- The '''sine-Gordon equation''' ...ontinuum of pendula hanging from a wire. It is a [[completely integrable]] equation, and has many interesting solutions, including "breather" solutions.705 bytes (103 words) - 23:39, 22 January 2009
- ...equation on the line and inverse scattering for the nonlinear Schrodinger equation with a potential319 bytes (42 words) - 20:26, 6 July 2009
- ...operation which converts solutions of one equation to solutions of another equation. Usually one only works with transforms which are given explicitly by an a302 bytes (43 words) - 00:13, 15 August 2006
- ...ints of that functional. It can be computed formally by starting with the equation ...iltonian]] can (in principle, at least) be expressed as the Euler-Lagrange equation of a functional, and conversely Euler-Lagrange equations can in principle b657 bytes (95 words) - 20:14, 11 June 2007
- ...the presence of non-compact symmetries (or approximate symmetries) in the equation and in the functional, however this can sometimes be counteracted using [[c899 bytes (129 words) - 01:41, 15 August 2006
- The (hyperbolic) '''minimal surface equation''' takes the form ...n \times R</math> ). This is the Minkowski analogue of the minimal surface equation in Euclidean space, see [[Hp1994]].809 bytes (125 words) - 22:10, 10 April 2007
- ...lize the solutions to have frequency comparable to one. In that case, the equation acquires a rescaling parameter <math>\hbar</math>, and the limit is usually867 bytes (126 words) - 20:02, 5 August 2006
- ...forward in time. The model instance of a dissipative equation is the heat equation ...ispersive equations are often [[Hamiltonian]] and are the [[Euler-Lagrange equation]] for some [[variational problem]], dissipative equations are usually non-H869 bytes (123 words) - 01:19, 15 August 2006
- The (homogeneous) '''Airy equation''' is given by ...tions. One such problem is the reduced (or linearized) [[Korteweg-de Vries equation]]1 KB (214 words) - 22:11, 19 January 2018
- ...asymptotically as <math>t \to \pm \infty</math> to solutions of the linear equation. Scattering can be formalized by rigorous results establishing the existen394 bytes (54 words) - 21:11, 9 August 2006
- The (defocusing) '''modified Korteweg-de Vries (mKdV) equation''' is ...roughly equivalent to the <span class="SpellE">H^k</span> norm of u. This equation has been studied [[modified Korteweg-de Vries on R|on the line]], [[modifie987 bytes (149 words) - 07:41, 31 July 2006
- #REDIRECT [[Hartree equation]]30 bytes (3 words) - 04:29, 29 July 2006
- '''Liouville's equation''' It is a limiting case of the [[Sine-Gordon|sinh-gordon equation]].2 KB (294 words) - 10:16, 27 January 2009
- #REDIRECT [[Kadomtsev-Petviashvili equation]]45 bytes (3 words) - 04:59, 27 July 2006
- #REDIRECT [[Semilinear Schrodinger equation]]45 bytes (4 words) - 03:05, 29 July 2006
- #REDIRECT [[Semilinear Schrodinger equation]]45 bytes (4 words) - 04:02, 29 July 2006
- #REDIRECT [[GKdV-3 equation]]29 bytes (3 words) - 20:06, 9 August 2006
- #REDIRECT [[GKdV-4 equation]]29 bytes (3 words) - 20:06, 9 August 2006
- #REDIRECT [[Benjamin-Ono equation]]35 bytes (3 words) - 18:38, 10 August 2006
- #REDIRECT [[Sine-Gordon equation]]34 bytes (3 words) - 15:50, 5 May 2007
- The '''free Schrodinger equation''' ...dimensions. One can also consider the inhomogeneous (forced) Schrodinger equation2 KB (307 words) - 07:45, 15 June 2009
- A '''quasilinear equation''' is an equation of the form ...nontrivial) in the top order terms <math>D^k u</math>. Thus a quasilinear equation takes the schematic form907 bytes (155 words) - 15:35, 12 December 2007
- #REDIRECT [[Korteweg-de Vries equation]]40 bytes (4 words) - 06:24, 31 July 2006
- #REDIRECT [[Ginsberg-Landau-Schrodinger equation]]50 bytes (3 words) - 14:24, 9 August 2006
- #REDIRECT [[Ginzburg-Landau-Schrodinger equation]]50 bytes (3 words) - 06:36, 16 January 2007
- #REDIRECT [[Generalized Korteweg-de Vries equation]]52 bytes (5 words) - 06:14, 31 July 2006
- #REDIRECT [[Derivative non-linear Schrodinger equation]]56 bytes (5 words) - 07:27, 31 July 2006
- #REDIRECT [[modified Korteweg-de Vries equation]]49 bytes (5 words) - 07:40, 31 July 2006
- #REDIRECT [[Generalized Korteweg-de Vries equation]]52 bytes (5 words) - 20:05, 9 August 2006
- #REDIRECT [[Korteweg-de Vries equation on T]]45 bytes (6 words) - 05:00, 8 August 2006
- #REDIRECT [[Korteweg-de Vries equation on R]]45 bytes (6 words) - 05:02, 8 August 2006
- ...larization method''' or '''viscosity method''') constructs solutions to an equation by the following general procedure: # Regularize the equation somehow (for instance, by adding penalty terms to the Hamiltonian or Lagran1 KB (174 words) - 01:03, 15 August 2006
- B. Birnir, G. Ponce, L. Vega, ''The local ill-posedness of the modified KdV equation. Annales de l'I. H. P.'', '''13''' (1996), 529-535. [http://archive.numdam. ...posedness for the initial value problem for the modified Korteweg-de Vries equation'', preprint.462 bytes (63 words) - 22:52, 26 February 2010
- #REDIRECT [[Korteweg-de Vries equation on the half-line]]57 bytes (7 words) - 20:03, 9 August 2006
- ...linear wave equation]] and a [[Schrodinger equations|nonlinear Schrodinger equation]]. The main examples are: ** The [[magnetic Zakharov equation]] (formed by adding a magnetic field to the Zakharov system)748 bytes (101 words) - 20:24, 9 August 2006
- ...nal equation; perturbation theory (and additional smoothing effects of the equation) is then used to convert the approximate solution to an exact one. The per ...ightly different method of mollifying the high frequency components of the equation and analyzing the energy of the resulting field, rather than1 KB (182 words) - 06:40, 16 January 2007
- The '''Korteweg-de Vries (KdV) equation''' is The equation is [[completely integrable]], and has infinitely many conserved quantities.2 KB (307 words) - 16:10, 14 September 2006
- When the linear equation is constant-coefficient then the Fourier transform is an excellent tool for ...be extracted by considering the [[small amplitude limit]] of the original equation.1 KB (145 words) - 04:29, 8 August 2006
- ...ve or a solution independent of the spatial variables) which collapses the equation to an ODE. This is a special case of the method of separation of variables ...p solutions to various equations such as the focusing [[NLW|nonlinear wave equation]].450 bytes (72 words) - 05:02, 4 August 2006
- H. Smith, Small data scattering for a nonlinearly damped wave equation, preprint.109 bytes (14 words) - 17:40, 8 August 2006
- ...al, F. Linares, ''Some properties for a higher order nonlinear Schrodinger equation'', Preprint.132 bytes (16 words) - 01:35, 1 August 2006
- ....org/abs/math.AP/0311227 Instability of the periodic nonlinear Schrodinger equation]'', preprint.173 bytes (24 words) - 01:48, 1 August 2006
- ...f which are perturbations of one sort or another of the [[free Schrodinger equation]]. One general class of such equations takes the form ...n derivatives then we refer to this equation as a [[semilinear Schrodinger equation]] (NLS). These equations (particularly the [[cubic NLS]]) arise as model eq4 KB (615 words) - 03:44, 8 February 2011
- H. Smith, D. Tataru, Counterexamples to Strichartz estimates for the wave equation with nonsmooth coefficients, preprint.149 bytes (18 words) - 17:37, 8 August 2006
- B. Yordanov, Blow-up for the one-dimensional Klein-Gordon equation with a cubic non-linearity, preprint.132 bytes (15 words) - 15:55, 8 August 2006
- I. Rodnianski, T. Tao, Global regularity for the Maxwell-Klein-Gordon equation in high dimensions, preprint.136 bytes (16 words) - 17:59, 8 August 2006
- ...Vries on R|complex mKdV]], and a cubic nonlinear [[Airy equation]]. This equation is a general model for propagation of pulses in an optical fiber [[Kod1985]1 KB (197 words) - 12:45, 11 July 2007
- ...[[Schrodinger maps|Schrodinger maps]], [[Cubic DNLS on R|DNLS]], [[Hartree equation|Hartree]], etc.) ...odified Korteweg-de Vries equation|mKdV]], [[Generalized Korteweg-de Vries equation|gKdV]], etc.)2 KB (321 words) - 18:34, 10 June 2018
- ...may be only a distribution) is then the '''fundamental solution''' of the equation. Another way of viewing this is that the fundamental solution is the solut589 bytes (90 words) - 04:32, 10 August 2006
- C.D. Sogge, Estimates for the Dirichlet-wave equation and applications to nonlinear wave equations, preprint.137 bytes (17 words) - 17:35, 8 August 2006
- The '''low-frequency limit''' of an equation concerns the behaviour of the equation for solutions which have very low frequency <math>|\xi| \to 0</math>.672 bytes (99 words) - 20:05, 5 August 2006
- ...foschi/t.pdf On the regularity of multilinear forms associated to the wave equation]'', PhD Thesis, January 2000.180 bytes (28 words) - 02:45, 1 August 2006
- ...estimates in hyperbolic space and global existence for the semilinear wave equation, TAMS 353 (2001), 785-807.160 bytes (19 words) - 16:22, 8 August 2006
- ...B. Zhang, X. Zhang, ''A note on self-similar solutions of the Schrodinger equation'', preprint.188 bytes (26 words) - 17:48, 8 August 2006
- Solutions to the [[Airy equation]] and [[KdV equations|its perturbations]] are either estimated in mixed spa ...nger equation]] in [[Bo1993b]], although the analogous spaces for the wave equation appeared earlier [[RaRe1982]], [[Be1983]] in the context of [[propagation o1 KB (189 words) - 19:58, 14 August 2006
- | title = On the Korteweg-de Vries equation192 bytes (20 words) - 23:04, 26 May 2009
- | title = Periodic nonlinear Schrödinger equation and invariant measures213 bytes (22 words) - 20:53, 7 May 2009
- | title = Inverse scattering for the nonlinear Schrödinger equation209 bytes (22 words) - 20:24, 6 July 2009
- | title = Solutions of the wave equation with localized energy204 bytes (22 words) - 23:46, 15 June 2009
- C. Sulem, P. Sulem, The nonlinear Schrodinger equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences 139, Spring178 bytes (20 words) - 16:36, 8 August 2006
- A global or integral '''conservation law''' for an evolution equation is any quantity ''Q(t)'' depending on the value of all the fields at time ' ...quantities to symmetries of the underlying equation, in the case that the equation is [[Hamiltonian]] or Lagrangian.2 KB (275 words) - 18:47, 15 September 2006
- | title = On well-posedness of the forced nonlinear Schrödinger equation200 bytes (23 words) - 02:11, 10 May 2009
- | title = Blow up and scattering in the nonlinear Schrödinger equation212 bytes (24 words) - 00:22, 11 May 2009
- The '''small amplitude limit''' for a nonlinear equation arises when considering initial position ...s, the small amplitude limit is usually just the linear counterpart of the equation; however when analyzing long times (e.g. times comparable to <math>1/\epsil724 bytes (111 words) - 05:23, 2 August 2006
- | title = Periodic nonlinear Schrödinger equation and invariant measures228 bytes (24 words) - 20:58, 7 May 2009
- | title = Decay estimate for the wave equation with potential218 bytes (24 words) - 19:36, 17 May 2009
- | title = On the derivative nonlinear Schrödinger equation212 bytes (24 words) - 00:21, 24 May 2009
- | title = Global solvability of the derivative nonlinear Schrodinger equation219 bytes (23 words) - 21:19, 5 June 2009
- | title = Unique continuation for the KdV equation201 bytes (23 words) - 21:10, 9 July 2009
- | title = Well-posedness for the Kadomtsev-Petviashvili II equation219 bytes (22 words) - 01:13, 30 June 2009
- | title = Regularity of solutions to the Schrödinger equation212 bytes (25 words) - 00:39, 21 June 2009
- | title = Scattering Theory for Nonlinear Klein-Gordon Equation with Sobolev Critical Power232 bytes (24 words) - 17:52, 10 June 2009
- ...n ODE of the form <math>-c f + {\frac{d^2}{dx^2}} f + f^2 = 0</math>. This equation can be explicitly solved in terms of the hyperbolic secant function. Other ...uations, and it is suspected (but unproven for any focusing non-integrable equation) that solutions for such equations, if they do not blow up, should generica2 KB (236 words) - 14:24, 24 August 2009
- | title = An example of blowup at infinity for a quasilinear wave equation228 bytes (26 words) - 00:24, 11 July 2009
- | title = Semiclassical limit of the nonlinear Schrödinger equation in small time225 bytes (25 words) - 00:54, 22 May 2009
- | title = On the generalized Benjamin-Ono equation211 bytes (23 words) - 18:26, 30 May 2009
- | title = On a nonlinear dispersive equation with time-dependent coefficients225 bytes (24 words) - 21:02, 5 June 2009
- | title = Global low-regularity solutions for Kadomtsev-Petviashvili equation.231 bytes (22 words) - 18:02, 4 July 2009
- | title = On global existence and scattering for the wave maps equation210 bytes (24 words) - 17:54, 4 July 2009
- | title = The Korteweg-de Vries equation: A survey of results207 bytes (23 words) - 22:42, 8 June 2009
- | title = Uniqueness of weak solution of the KdV equation195 bytes (22 words) - 21:16, 9 July 2009
- | title = Invariant measures for the 2d-defocussing nonlinear Schrödinger equation239 bytes (25 words) - 21:03, 7 May 2009
- | title = Periodic Korteweg de Vries equation with measures as initial data233 bytes (27 words) - 23:52, 18 January 2018
- ...Resonance and long time existence for the quadratic nonlinear Schrödinger equation231 bytes (26 words) - 01:07, 13 May 2009
- | title = On the initial value problem for a nonlinear Schrödinger equation223 bytes (29 words) - 01:34, 15 May 2009
- | title = Profile decomposition for the wave equation outside a convex obstacle230 bytes (26 words) - 00:49, 17 May 2009
- | title = Global solution to the two-dimensional Klein-Gordon equation225 bytes (23 words) - 19:35, 17 May 2009
- | title = Regularity for the wave equation with a critical nonlinearity, Comm232 bytes (26 words) - 00:17, 22 May 2009
- | title = Cauchy problem for a generalized nonlinear dispersive equation218 bytes (24 words) - 00:50, 23 May 2009
- | title = Scattering and self-similar solutions for the nonlinear wave equation251 bytes (25 words) - 23:07, 24 May 2009
- | title = Multivortex solutions of a two-dimensional nonlinear wave equation232 bytes (24 words) - 21:02, 25 May 2009
- | title = Multidimensional inverse scattering for the nonlinear Klein-Gordon equation with a potential240 bytes (25 words) - 20:34, 6 July 2009
- | title = Ferromagnetic chain equation from a closed Riemannian manifold into <math>S^2</math>241 bytes (29 words) - 20:39, 6 July 2009
- | title = Rough solutions for the wave maps equation (English summary)235 bytes (25 words) - 20:25, 29 June 2009
- | title = On the local regularity of Kadomtsev-Petviashvili-II equation223 bytes (23 words) - 01:19, 30 June 2009
- | title = Global well-posedness for the Kadomtsev-Petviashvili II equation239 bytes (24 words) - 01:12, 30 June 2009
- | title = Quadratic forms for a <math>2-D</math> semilinear Schrödinger equation232 bytes (27 words) - 16:55, 22 June 2009
- | title = Remarks on the Korteweg-de Vries equation210 bytes (25 words) - 19:11, 16 June 2009
- | title = Nonlinear Klein-Gordon equation on Schwarzschild-like metrics232 bytes (24 words) - 18:22, 10 June 2009
- | title = Inhomogeneous Strichartz estimates for the Schrödinger equation239 bytes (26 words) - 19:04, 9 March 2011
- | title = Globalizing estimates for the periodic KPI equation216 bytes (23 words) - 00:27, 13 May 2009
- | title = On a generalized Kadomtsev-Petviashvili equation223 bytes (22 words) - 01:22, 30 June 2009
- .... Tao, ''Global well-posedness for large data for the Maxwell-Klein-Gordon equation below the energy norm'', in preparation.218 bytes (29 words) - 01:31, 3 August 2006
- ==Maxwell-Dirac equation== [More info on this equation would be greatly appreciated. - Ed.]5 KB (851 words) - 11:32, 26 March 2011
- C. Wilcox, The initial boundary-value problem for the wave equation in an exterior domain with spherical boundary, Not. Amer. Math. Soc. 6 (195174 bytes (23 words) - 16:00, 8 August 2006
- | title = The periodic cubic Schrödinger equation232 bytes (27 words) - 23:00, 4 May 2009
- ...Nonexistence of asymptotically free solutions for a nonlinear Schrodinger equation245 bytes (27 words) - 20:37, 5 May 2009
- | title = Decay and regularity for the Schrödinger equation226 bytes (26 words) - 17:17, 6 May 2009
- | title = A higher-order nonlinear Schrödinger equation with variable coefficients276 bytes (27 words) - 01:08, 14 May 2009
- | title = Rapidly decaying solutions of the nonlinear Schrödinger equation245 bytes (28 words) - 00:33, 11 May 2009
- | title = On solutions of the derivative nonlinear Schrödinger equation II237 bytes (27 words) - 00:33, 17 May 2009
- | title = Modified wave operators for the derivative nonlinear Schrödinger equation242 bytes (27 words) - 00:25, 24 May 2009
- | title = On the Maxwell-Klein-Gordon equation with finite energy229 bytes (25 words) - 01:37, 29 May 2009
- | title = The Cauchy problem for a third order nonlinear Schrödinger equation233 bytes (28 words) - 20:59, 5 June 2009
- ...itle = On the asymptotic behavior of solutions for a nonlinear Schrodinger equation229 bytes (27 words) - 00:40, 7 June 2009
- | title = Remark on the local ill-posedness for KdV equation222 bytes (26 words) - 18:01, 4 July 2009
- | title = On solutions to the derivative nonlinear Schrödinger equation II234 bytes (27 words) - 22:33, 1 July 2009
- | title = On the Korteweg-de Vries equation: Existence and uniqueness227 bytes (26 words) - 00:37, 21 June 2009
- | title = Exponential decay of solutions to the wave equation227 bytes (27 words) - 00:31, 10 June 2009
- | title = Time decay for the non-linear Klein-Gordon equation219 bytes (25 words) - 17:36, 10 June 2009
- ...itle = Asymptotically free solutions for short range nonlinear Schrödinger equation246 bytes (27 words) - 17:54, 10 June 2009
- | title = Dispersive estimates and the 2D cubic NLS equation (English summary)241 bytes (26 words) - 19:03, 13 June 2009
- | title = Scattering for the quartic generalized Korteweg-de Vries equation235 bytes (24 words) - 18:05, 27 June 2009
- | title = The Cauchy problem for the Kadomtsev-Petviashvili equation224 bytes (24 words) - 02:00, 16 May 2009
- | title = Nonlinear Schrödinger equation245 bytes (26 words) - 23:14, 26 May 2009
- | title = The nonrelativistic limit of the nonlinear Klein-Gordon equation232 bytes (23 words) - 01:08, 7 June 2009
- | title = Local decay of scattering solutions to Schrödinger's equation210 bytes (23 words) - 23:36, 15 June 2009
- J. Ginibre, ''Equations d'Evolution Semilinearies: L'Equation de Schrodinger Non-lineaire'', Cours de DEA, Universite de Paris-Sud, Paris169 bytes (18 words) - 03:02, 1 August 2006
- ...equation]] with non-local nonlinearity (somewhat similar to the [[Hartree equation]]), and can be treated by [[Strichartz estimates]] [[GhSau1990]]; for <math ...m. A particularly interesting solution of the focusing Davey-Stewartson II equation was found by [[Oza1992]] and is2 KB (317 words) - 13:41, 29 May 2012
- | title = On the Cauchy problem for the Kadomtsev-Petviashvili equation226 bytes (24 words) - 20:46, 7 May 2009
- | title = The scattering transform for the Benjamin-Ono equation232 bytes (24 words) - 00:28, 14 May 2009
- | title = Generalized Strichartz Inequalities for the Wave Equation224 bytes (25 words) - 20:10, 18 May 2009
- | title = On Lebesgue space estimates for the wave equation217 bytes (25 words) - 22:56, 24 May 2009
- | title = On the Cauchy problem for the Benjamin-Ono equation216 bytes (26 words) - 21:06, 25 May 2009
- | title = The Cauchy problem for a third order nonlinear Schrödinger equation237 bytes (27 words) - 21:01, 5 June 2009
- | title = Decay and scattering of solutions of nonlinear Schrödinger equation227 bytes (26 words) - 00:54, 7 June 2009
- ...l well-posedness of the initial value problem associated with the Benjamin equation259 bytes (29 words) - 21:31, 5 June 2009
- ...itle = Decay and scattering of small solutions of a generalized Boussinesq equation222 bytes (25 words) - 00:46, 7 June 2009
- ...Strong instability of solitary wave solutions of a generalized Boussinesq equation230 bytes (25 words) - 00:47, 7 June 2009
- | title = Well-posedness for the Schrödinger equation with the derivative non-linearity238 bytes (26 words) - 01:11, 30 June 2009
- | title = On the global well-posedness of the Benjamin-Ono equation232 bytes (24 words) - 23:28, 15 June 2009
- | title = On the (generalized) Korteweg-de Vries equation224 bytes (26 words) - 23:30, 29 May 2009
- | title = Optimal bounds on positive blowup solutions for a semilinear wave equation239 bytes (27 words) - 23:52, 4 May 2009
- | title = Solution of the forced nonlinear Schrödinger (NLS) equation using PDE techniques242 bytes (29 words) - 00:49, 14 May 2009
- | title = Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow241 bytes (28 words) - 01:33, 15 May 2009
- | title = Regularity and asymptotic behaviour of the wave equation with a critical non-linearity246 bytes (27 words) - 00:08, 22 May 2009
- | title = Solutions to the Korteweg-de Vries equation with irregular initial profile259 bytes (27 words) - 16:24, 4 June 2009
- | title = Instability and blowup of solutions to a generalized Boussinesq equation235 bytes (27 words) - 00:44, 7 June 2009
- ...e = Instability of solitons for the critical generalized Korteweg de Vries equation238 bytes (27 words) - 00:18, 10 June 2009
- | title = Unique continuation properties of the nonlinear Schrödinger equation249 bytes (28 words) - 21:13, 9 July 2009
- | title = Regularity of solutions to the free Schrödinger equation with radial initial data246 bytes (29 words) - 18:13, 4 July 2009
- | title = Well-posedness for the higher order nonlinear Schrödinger equation (English summary)265 bytes (28 words) - 20:16, 29 June 2009
- | title = On the critical semilinear wave equation outside convex obstacles239 bytes (29 words) - 00:53, 21 June 2009
- | title = Global well-posedness for the KP-I equation229 bytes (26 words) - 23:53, 9 June 2009
- ...tle = On blowup for the pseudo-conformally invariant nonlinear Schrödinger equation. II.244 bytes (27 words) - 00:41, 13 June 2009
- | title = On self-similar solutions, well-posedness and the conformal wave equation (English summary)270 bytes (27 words) - 19:05, 13 June 2009
- | title = The Cauchy problem for the Korteweg-de Vries equation218 bytes (25 words) - 23:06, 26 May 2009
- | title = The nonlinear Schrödinger equation on the semi-infinite line227 bytes (26 words) - 20:53, 8 May 2009
- | title = On the Cauchy problem for the Korteweg de Vries equation220 bytes (26 words) - 00:07, 11 May 2009
- | title = More self-similar solutions of the nonlinear Schrödinger equation254 bytes (29 words) - 00:36, 11 May 2009
- | title = A construction of the fundamental solution for the Schrödinger equation232 bytes (27 words) - 00:34, 17 May 2009
- | title = An improved local well-posedness result for the modified KdV equation242 bytes (27 words) - 00:47, 22 May 2009
- | title = Nonlinear wave interactions for the Benjamin-Ono equation240 bytes (24 words) - 18:56, 30 May 2009
- | title = Smoothing properties of some weak solutions of the Benjamin-Ono equation248 bytes (27 words) - 01:21, 30 June 2009
- | title = Determination of the blowup rate for the semilinear wave equation228 bytes (26 words) - 20:40, 8 June 2009
- | title = Concerning the Wave equation on Asymptotically Euclidean Manifolds235 bytes (26 words) - 21:09, 9 March 2011
- | title = Global existence for the nonresonant Schrödinger equation in two space dimensions249 bytes (29 words) - 00:26, 14 May 2009
- | title = Global well-posedness of the KdV equation below L^2243 bytes (28 words) - 01:05, 13 May 2009
- | title = On the wave equation with large rough potential240 bytes (27 words) - 00:51, 15 May 2009
- | title = A priori estimates for the 2-D wave equation239 bytes (27 words) - 02:13, 16 May 2009
- | title = On global solutions to a defocusing semi-linear wave equation246 bytes (27 words) - 00:50, 17 May 2009
- | title = Scattering theory for the Hartree type equation239 bytes (29 words) - 00:11, 24 May 2009
- ...cattering problem and asymptotics for a relativistic nonlinear Schrödinger equation272 bytes (31 words) - 00:16, 24 May 2009
- | title = Small data scattering and blow-up for the wave equation with a cubic convolution248 bytes (28 words) - 23:05, 24 May 2009
- | title = Global solutions for the KdV equation with unbounded data250 bytes (29 words) - 18:33, 30 May 2009
- | title = Nonlinear optics and supercritical wave equation. Hommage à Pascal Laubin247 bytes (27 words) - 21:26, 5 June 2009
- ...ability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation254 bytes (26 words) - 00:48, 7 June 2009
- ...averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation263 bytes (27 words) - 21:30, 23 June 2009
- | title = Strichartz estimates for the Schrödinger equation with radial data (English summary)281 bytes (30 words) - 17:04, 22 June 2009
- | title = <math>L^p</math> regularity for the wave equation with strictly convex obstacles248 bytes (32 words) - 00:50, 21 June 2009
- ...le = On blow-up for the pseudo-conformally invariant nonlinear Schrödinger equation253 bytes (27 words) - 00:39, 13 June 2009
- | title = On the Cauchy problem in Besov spaces for a nonlinear Schrödinger equation243 bytes (29 words) - 23:19, 15 June 2009
- | title = The nonrelativistic limit of the nonlinear Klein-Gordon equation225 bytes (23 words) - 18:25, 10 June 2009
- | title = Nonlinear Schrödinger equation with harmonic potential238 bytes (26 words) - 00:13, 21 June 2009
- ...erivative nonlinear wave equation''' is a [[DNLW|derivative nonlinear wave equation]] of with the schematic form This equation has the same scaling as [[cubic NLW]], but is more difficult technically be3 KB (540 words) - 23:10, 14 August 2006
- | title = The Cauchy problem for the critical nonlinear Schrödinger equation244 bytes (29 words) - 12:23, 14 October 2010
- | title = The nonlinear Schrödinger equation on the interval241 bytes (29 words) - 00:26, 17 May 2009
- | title = The global Cauchy problem for the nonlinear Klein-Gordon equation228 bytes (26 words) - 20:00, 18 May 2009
- ...linear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation245 bytes (27 words) - 00:51, 7 June 2009
- | title = On the equation <math> u = |\nabla u|^2</math> in <math>R^{5+1}</math>234 bytes (28 words) - 22:38, 1 July 2009
- | title = A parametrix construction for wave equation with <math>C^{1,1}</math> coefficients253 bytes (27 words) - 00:44, 21 June 2009
- | title = "Mass concentration" phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity268 bytes (29 words) - 16:11, 12 June 2009
- ...le = Unique continuation property for a higher order nonlinear Schrödinger equation267 bytes (30 words) - 01:09, 14 May 2009
- | title = Stability of stationary waves for a quasilinear Schrödinger equation in space dimension 2258 bytes (29 words) - 00:31, 14 May 2009
- ...itle = Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation253 bytes (27 words) - 01:52, 16 May 2009
- | title = On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem269 bytes (30 words) - 00:32, 17 May 2009
- | title = Existence of self-similar blow-up solutions for Zakharov equation in dimension two I246 bytes (28 words) - 19:40, 17 May 2009
- | title = Nonlinear small data scattering for the wave equation in <math>R^{4+1}</math>242 bytes (29 words) - 23:12, 24 May 2009
- ...itle = Regularizing effects for the linearized Kadomtsev-Petviashvili (KP) equation266 bytes (28 words) - 21:16, 25 May 2009
- | title = A bilinear estimate with applications to the KdV equation239 bytes (30 words) - 18:31, 30 May 2009
- | title = Local solutions of the Kadomtsev-Petviashvili equation237 bytes (27 words) - 18:04, 4 July 2009
- | title = On solutions to the derivative nonlinear Schrödinger equation. Existence and Uniqueness theorem267 bytes (30 words) - 22:32, 1 July 2009
- | title = From non-linear Klein-Gordon equation to a system of coupled Schrödinger equations256 bytes (29 words) - 01:16, 7 June 2009
- ...rmination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power265 bytes (31 words) - 20:26, 8 June 2009
- ...math>L^2</math> minimal blowup solutions of critical nonlinear Schrödinger equation272 bytes (31 words) - 20:27, 8 June 2009
- ...lity of blowup profile for <math>L^2</math> critical nonlinear Schrödinger equation266 bytes (31 words) - 20:36, 8 June 2009
- ...= Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation I256 bytes (30 words) - 21:05, 17 June 2009
- | title = Blowup solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary conditions276 bytes (31 words) - 00:47, 13 June 2009
- | title = Blow-up of <math>H^1</math> solution for the nonlinear Schrödinger equation259 bytes (30 words) - 00:49, 13 June 2009
- | title = Some Remarks on Strichartz Estimates for Homogeneous Wave Equation263 bytes (28 words) - 19:29, 8 March 2011
- | title = Decay for solutions of the exterior problem for the wave equation220 bytes (26 words) - 17:38, 10 June 2009
- {{equation | equation = <math>iu_t + u_{xx} = \pm |u|^2 u</math>2 KB (260 words) - 05:06, 8 August 2006
- | title = Decay of solutions of the wave equation outside of convex obstacles237 bytes (29 words) - 17:40, 10 June 2009
- {{equation | equation = <math>u_t + u_{xxx} = \pm u^3 u_x</math>2 KB (376 words) - 22:26, 4 March 2007
- | title = Microlocal dispersive smoothing for the Schrodinger equation250 bytes (28 words) - 00:47, 14 May 2009
- | title = On the local well-posedness of the Benjamin-Ono equation in <math>Hs(R)</math>242 bytes (29 words) - 18:51, 30 May 2009
- .../math> spaces and unique continuation for solutions to the semilinear wave equation253 bytes (31 words) - 22:35, 1 July 2009
- | title = <math>L^2</math> stability of solitons for the KdV equation238 bytes (29 words) - 20:39, 8 June 2009
- | title = Local well-posedness for the Maxwell-Schrödinger equation (English summary)268 bytes (27 words) - 15:56, 12 June 2009
- ...tic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power260 bytes (30 words) - 16:21, 12 June 2009
- | title = On the Cauchy problem for the n-dimensional wave equation230 bytes (27 words) - 12:14, 23 June 2009
- | title = On the Cauchy problem for the n-dimensional wave equation230 bytes (27 words) - 12:18, 23 June 2009
- | title = Asymptotic solutions of the Korteweg-deVries equation252 bytes (27 words) - 22:38, 4 May 2009
- ...r the minimal periodic blow-up solutions of critical nonlinear Schrödinger equation291 bytes (30 words) - 23:49, 4 May 2009
- | title = Similarity solutions of the generalized Korteweg-de Vries equation269 bytes (29 words) - 20:28, 7 May 2009
- | title = A method for solving the Korteweg-de Vries equation240 bytes (29 words) - 19:22, 17 May 2009
- ...ce of small solutions to the genrealized derivative non-linear Schrödinger equation274 bytes (30 words) - 01:10, 23 May 2009
- ...= Large time asymptotics of solutions to the generalized Korteweg-de Vries equation258 bytes (30 words) - 00:02, 24 May 2009
- ...itle = Large time asymptotics of solutions to the generalized Benjamin-Ono equation261 bytes (30 words) - 00:07, 24 May 2009
- | title = A regularity theorem for the Kadomtsev-Petviashvili equation with periodic boundary conditions277 bytes (30 words) - 21:13, 25 May 2009
- | title = Local solution for the Kadomtsev-Petviashvili equation in <math>R^2</math>261 bytes (30 words) - 21:15, 25 May 2009
- | title = Quadratic forms for the 1-D semilinear Schrodinger equation249 bytes (29 words) - 18:31, 30 May 2009
- | title = Sharp local well-posedness results for the nonlinear wave equation. (English summary)272 bytes (29 words) - 00:55, 21 June 2009
- ...f solutions with exactly <math>k</math> blow-up points for the Schrödinger equation with critical non-linearity288 bytes (34 words) - 20:25, 8 June 2009
- | title = Well-posedness results for the generalized Benjamin-Ono equation with small initial data266 bytes (30 words) - 23:49, 9 June 2009
- | title = Low regularity stability of solitons for the KdV equation254 bytes (29 words) - 23:50, 15 June 2009
- ...rgence theorems for the pseudo-conformally invariant nonlinear Schrödinger equation269 bytes (30 words) - 16:08, 12 June 2009
- ...tle = Asymptotic profiles of blowup solutions to the nonlinear Schrödinger equation with critical power nonlinearity276 bytes (32 words) - 16:13, 12 June 2009
- ...e = Non-linear small data scattering for the generalized Korteweg-de Vries equation249 bytes (28 words) - 23:30, 15 June 2009
- | title = Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions260 bytes (29 words) - 17:04, 27 June 2009
- | title = On L^p - L^p' estimates for the wave equation200 bytes (26 words) - 20:28, 8 May 2009
- ...e decay of solutions to the exterior initial-boundary problem for the wave equation239 bytes (28 words) - 00:30, 10 June 2009
- | title = Inverse scattering transform for the time-dependent Schrödinger equation with application to the KP-I equations261 bytes (29 words) - 00:14, 11 July 2009
- | title = The Korteweg-de Vries equation, posed in a quarter plane233 bytes (28 words) - 20:45, 7 May 2009
- | title = An instability property of the nonlinear Schrodinger equation on S^d239 bytes (29 words) - 23:45, 10 May 2009
- | title = An instability property of the nonlinear Schrodinger equation on S^d239 bytes (29 words) - 23:56, 10 May 2009
- | title = Global well-posedness for the critical generalized KdV equation264 bytes (29 words) - 02:19, 16 May 2009
- | title = Uniqueness of solutions for the generalized Korteweg-de Vries equation249 bytes (28 words) - 19:56, 17 May 2009
- ...harp upper bound on the blowup rate for the critical nonlinear Schrödinger equation247 bytes (30 words) - 20:34, 8 June 2009
- | title = On the bound states of the nonlinear Schrödinger equation with a linear potential249 bytes (31 words) - 00:04, 16 June 2009
- ...nequalities and uniqueness of weak solutions for the nonlinear Schrödinger equation260 bytes (30 words) - 00:45, 13 June 2009
- ...rsal symmetry, this symmetry manifests itself differently from equation to equation: * For [[KdV-type equations]] and the [[Benjamin-Ono equation]], the time-reversal of a solution <math>u(t,x)</math> is usually <math>u(-2 KB (321 words) - 17:17, 15 August 2006
- | title = On the inverse scattering transform for the Ishimori equation243 bytes (27 words) - 19:01, 30 May 2009
- | title = Scattering for the nonlinear Schrödinger equation: states that are close to a solitone269 bytes (33 words) - 00:09, 11 May 2009
- | title = Low regularity solutions of the Kadomtsev-Petviashvili-I equation278 bytes (28 words) - 00:56, 13 May 2009
- | title = Low regularity solutions for the Kadomtsev-Petviashvili I equation279 bytes (29 words) - 01:00, 13 May 2009
- ...xistence and uniqueness of solutions for the generalized Korteweg de Vries equation257 bytes (31 words) - 19:57, 17 May 2009
- | title = Global solution for the Kadomtsev-Petviashvili equation (KPII) in anisotropic Sobolev spaces of negative index266 bytes (30 words) - 21:12, 25 May 2009
- | title = An elementary proof of the blowup for a semilinear wave equation in high space dimensions247 bytes (31 words) - 21:45, 25 May 2009
- | title = Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity295 bytes (35 words) - 23:58, 26 May 2009
- | title = Global existence for a quasilinear wave equation outside of star-shaped domains263 bytes (31 words) - 00:09, 27 May 2009
- | title = Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds278 bytes (31 words) - 17:01, 4 June 2009
- | title = Mixed-norm estimates for the Klein-Gordon equation258 bytes (28 words) - 00:08, 10 June 2009
- | title = The Cauchy problem for non-linear Klein-Gordon equation and applications (English summary)268 bytes (29 words) - 01:06, 21 June 2009
- | title = Two points blow up in solutions to the nonlinear Schrödinger equation with quartic potential on <math>R</math>268 bytes (35 words) - 16:19, 12 June 2009
- ...mmetry is the same concept as a [[transform]], except that the transformed equation is the same as the old. Note that any given equation will typically only enjoy a subset of the symmetries on this list.2 KB (203 words) - 17:22, 15 August 2006
- A semilinear equation is said to have ''null structure'' if the [[resonant]] component of the non The only bilinear forms in first derivatives of scalar solutions to the wave equation which have null structure are linear combinations of the above forms. Howe3 KB (460 words) - 17:19, 2 June 2010
- {{equation | equation = <math>iu_t + \Delta u = \pm |u|^2 u</math>1 KB (209 words) - 06:24, 21 July 2007
- | title = White noise driven Korteweg-de Vries equation236 bytes (27 words) - 20:55, 5 May 2009
- | title = On the asymptotic behaviour of solutions of the wave equation233 bytes (28 words) - 20:41, 8 May 2009
- | title = Decay estimates for the critical semilinear wave equation259 bytes (31 words) - 20:22, 5 May 2009
- ...ity of solitary-wave solutions of the 3-dimensional Kadomtsev-Petviashvili equation254 bytes (27 words) - 03:15, 10 May 2009
- | title = Solutions of the Korteweg-de Vries equation, in fractional order Sobolev spaces243 bytes (29 words) - 20:20, 7 May 2009
- ...lobal existence of small solutions to a relativistic nonlinear Schrödinger equation261 bytes (32 words) - 20:57, 5 May 2009
- | title = Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds.271 bytes (31 words) - 23:48, 10 May 2009
- | title = Global well-posedness for the modified Korteweg-de Vries equation269 bytes (29 words) - 02:20, 16 May 2009
- | title = On the point-particle (Newtonian) limit of the non-linear Hartree equation242 bytes (28 words) - 00:21, 17 May 2009
- | title = The global Cauchy problem for the nonlinear Schrödinger equation revisited265 bytes (33 words) - 20:01, 18 May 2009
- ...hing properties and existence of solutions for the generalized Bejamin-Ono equation268 bytes (29 words) - 20:08, 18 May 2009
- | title = Well-posedness of the initial value problem for the KdV equation246 bytes (31 words) - 23:34, 29 May 2009
- | title = Inverse scattering on the line for the nonlinear Klein-Gordon equation with a potential253 bytes (30 words) - 20:33, 6 July 2009
- ...e = The null gauge condition and the one-dimensional nonlinear Schrödinger equation with cubic nonlinearity274 bytes (31 words) - 21:35, 1 July 2009
- | title = The Cauchy problem for the Korteweg-de Vries equation with measures as initial data250 bytes (30 words) - 21:31, 1 July 2009
- | title = Global well-posedness for the Benjamin-Ono equation in <math>H^1(R)</math>253 bytes (29 words) - 18:01, 27 June 2009
- ...tle = Formation of singularities in solutions of the nonlinear Schrödinger equation256 bytes (31 words) - 16:09, 12 June 2009
- | title = Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition269 bytes (31 words) - 01:20, 30 June 2009
- | title = Fully discrete Galerkin methods for the Korteweg-de Vries equation270 bytes (33 words) - 20:15, 7 May 2009
- ...= On the existence of weak solutions for a time-dependent nonlinear Dirac equation265 bytes (32 words) - 01:29, 15 May 2009
- ...ation properties of blow-up solutions and instability results for Zakharov equation in dimension two. II.275 bytes (31 words) - 19:39, 17 May 2009
- ...otic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited293 bytes (34 words) - 23:59, 23 May 2009
- | title = <math>H^m</math>-modified wave operator for nonlinear Hartree equation in the dimensions space <math>n \geq 2</math>278 bytes (35 words) - 22:48, 8 June 2009
- | title = On an estimate for the wave equation and applications to nonlinear problems (English summary)294 bytes (31 words) - 20:51, 17 June 2009
- | title = Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm252 bytes (29 words) - 16:54, 27 June 2009
- | title = The initial value problem for the derivative nonlinear Schrödinger equation in the energy space257 bytes (30 words) - 00:57, 23 May 2009
- ...Tzvetkov, ''An example of singular dynamics for the nonlinear Schrodinger equation on bounded domains, Proc. conference on hyperbolic PDE and related topics,257 bytes (35 words) - 01:04, 1 August 2006
- | title = An exact solution for a derivative nonlinear Schrödinger equation240 bytes (30 words) - 00:01, 27 May 2009
- | title = The generalized Korteweg-de Vries equation on the half line278 bytes (29 words) - 00:54, 13 May 2009
- | title = Blowup of small data solutions for a quasilinear wave equation in two space dimensions258 bytes (30 words) - 23:37, 4 May 2009
- | title = The initial-value problem for the Korteweg-de Vries equation257 bytes (30 words) - 20:21, 7 May 2009
- ...= Non existence of L^2-compact solutions of the Kadomtsev-Petviashvili II equation.254 bytes (29 words) - 21:00, 5 May 2009
- | title = A semilinear Dirac equation in H^s(R^3) for s > 1222 bytes (29 words) - 01:56, 16 May 2009
- | title = Scattering problem for the nonlinear wave equation in the finite energy and conformal charge space266 bytes (31 words) - 23:09, 24 May 2009
- | title = On the unique continuation of solutions to the generalized KdV equation255 bytes (31 words) - 23:25, 29 May 2009
- | title = Scattering for the forced non-linear Schrödinger equation with a potential on the half-line271 bytes (31 words) - 18:24, 4 July 2009
- | title = Global well-posedness for the Klein-Gordon equation below the energy norm274 bytes (31 words) - 22:56, 8 June 2009
- <center>'''Benjamin-Ono equation'''</center> The Benjamin-Ono equation (BO) [[Bj1967]], [[On1975]], which models one-dimensional internal waves in4 KB (707 words) - 18:49, 25 October 2008
- ...ice subsets and applications to nonlinear evolution equations. II. The KdV-equation307 bytes (33 words) - 20:43, 7 May 2009
- | title = Sur le temps d'existence pour l'equation de Klein-Gordon semi-lineaire en dimension 1254 bytes (28 words) - 00:55, 15 May 2009
- | title = The Korteweg-de Vries equation and its generalizations VI, Method for exact solutions267 bytes (32 words) - 19:24, 17 May 2009
- ...r large time of global solutions to the generalized Kadomtsev-Petviashvili equation286 bytes (34 words) - 00:15, 24 May 2009
- ...Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation285 bytes (33 words) - 01:13, 7 June 2009
- | title = Cross-constrained variational problem and nonlinear Schrödinger equation317 bytes (34 words) - 21:28, 9 July 2009
- ...Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation282 bytes (33 words) - 21:57, 8 March 2011
- ...l-posedness of the Cauchy problem for the generalized nonlinear dispersive equation267 bytes (31 words) - 23:32, 24 May 2009
- | title = The mKdV equation on the half-line240 bytes (29 words) - 21:11, 5 May 2009
- ...spersion of small amplitude solutions of the generalized Korteweg-de Vries equation258 bytes (29 words) - 00:56, 14 May 2009
- | title = Note on a modification to the nonlinear Schrödinger equation for application to deep water261 bytes (33 words) - 01:47, 16 May 2009
- ...= On the existence of self-similar solutions of the nonlinear Schrödinger equation with power nonlinearity between 1 and 2272 bytes (31 words) - 00:42, 17 May 2009
- | title = On the solvability of a mixed problem for a nonlinear equation of Schrodinger type241 bytes (30 words) - 18:08, 4 July 2009
- A '''semilinear''' equation is a PDE of the form A '''semilinear-with-derivatives''' equation is a PDE of the form706 bytes (113 words) - 15:41, 12 December 2007
- | title = A note on the nonlinear Schrödinger equation in weak L^p spaces.264 bytes (34 words) - 00:30, 11 May 2009
- | title = The emergence of solutions to the Korteweg-de Vries equation from arbitrary initial conditions269 bytes (32 words) - 01:51, 16 May 2009
- | title = The inverse scattering transform for the Benjamin-Ono equation: a pivot to multidimensional problems274 bytes (32 words) - 00:24, 17 May 2009
- | title = Scattering theory for the Schrödinger equation in some external time-dependent magnetic fields298 bytes (32 words) - 19:50, 18 May 2009
- | title = On the local and global well-posedness theory for the KP-I equation259 bytes (33 words) - 23:15, 29 May 2009
- ...all global solutions and the nonrelativistic limit for the nonlinear Dirac equation285 bytes (32 words) - 01:11, 7 June 2009
- | title = <math> L^p \rightarrow L^q</math> estimates for the Klein-Gordon equation264 bytes (33 words) - 00:10, 10 June 2009
- | title = Lower bounds for the blowup rate of solutions to the Zakharov equation in dimension 2252 bytes (31 words) - 20:29, 8 June 2009
- ...of blow-up solutions in the energy space for the critical generalized KdV equation256 bytes (31 words) - 20:31, 8 June 2009
- | title = Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle285 bytes (32 words) - 20:41, 8 June 2009
- ...The initial value problem for the <math>1-D</math> semilinear Schrödinger equation in Besov spaces277 bytes (35 words) - 00:27, 10 June 2009
- | title = Angular regularity and Strichartz estimates for the wave equation. With an appendix by Igor Rodnianski281 bytes (32 words) - 18:24, 9 March 2011
- | title = The equation Box<math> u = |u|^p</math> fror the critical value of p243 bytes (32 words) - 20:56, 17 June 2009
- | title = Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation290 bytes (34 words) - 22:59, 4 May 2009
- ...,\,2)</math> estimate for a maximal operator associated to the Schrödinger equation266 bytes (32 words) - 18:42, 30 May 2009
- | title = Inverse scattering for the nonlinear Schrödinger equation: reconstruction of the potential and the non-linearity282 bytes (32 words) - 20:29, 6 July 2009
- ...st global existence of solutions for the quadratic semilinear Klein-Gordon equation in one space dimension300 bytes (34 words) - 00:05, 10 June 2009
- | title = Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces.286 bytes (31 words) - 23:54, 10 May 2009
- | title = On solutions for the Kadomtsev-Petviashvili I equation270 bytes (28 words) - 00:58, 13 May 2009
- | title = The global Cauchy problem for the critical non-linear wave equation247 bytes (30 words) - 19:55, 17 May 2009
- | title = The forced non-linear Schrödinger equation with a potential on the half-line270 bytes (30 words) - 18:19, 4 July 2009
- ...^2</math> concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power282 bytes (33 words) - 21:32, 1 July 2009
- ...imates for operators with non-smooth co-efficients and the non-linear wave equation276 bytes (29 words) - 17:52, 4 July 2009
- | title = Asymptotics for large time for nonlinear Schrödinger equation. (English summary)292 bytes (33 words) - 16:00, 12 June 2009
- | title = A remark on long range scattering for the nonlinear Klein-Gordon equation270 bytes (31 words) - 21:15, 5 June 2009
- ...and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation276 bytes (32 words) - 20:37, 6 July 2009
- | title = The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension (English summary)308 bytes (32 words) - 20:32, 29 June 2009
- | title = Angular regularity and Strichartz estimates for the wave equation. With an appendix by Igor Rodnianski293 bytes (33 words) - 21:06, 23 June 2009
- ...time for <math>L^2</math> solutions of the critical nonlinear Schrodinger equation in 2D269 bytes (33 words) - 20:37, 8 June 2009
- | title = Modified wave operators for the Hartree equation with data, image and convergence in the same space. (English summary)295 bytes (35 words) - 17:45, 10 June 2009
- | title = Modified wave operators for the Hartree equation with data, image and convergence in the same space. II. (English summary)295 bytes (35 words) - 17:47, 10 June 2009
- | title = Limiting profiles of blowup solutions of the nonlinear Schrodinger equation with critical power nonlinearity286 bytes (34 words) - 16:14, 12 June 2009
- ...e = Dilation method and smoothing effects of solutions to the Benjamin-Ono equation280 bytes (34 words) - 01:09, 23 May 2009
- | title = On the long-time behavior of a generalized KdV equation244 bytes (31 words) - 00:33, 21 June 2009
- ...Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation267 bytes (29 words) - 23:54, 9 June 2009
- | title = <math>L^p</math> estimates for the wave equation and global existence for semilinear wave equations in exterior domains277 bytes (34 words) - 18:29, 10 June 2009
- | title = Decay of solutions to the wave equation to the exterior of several convex bodies254 bytes (31 words) - 21:00, 25 May 2009
- | title = A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain237 bytes (29 words) - 21:11, 9 July 2009
- | title = Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices294 bytes (34 words) - 01:05, 14 May 2009
- | title = The potential form of the fourth-order evolution equation for deep-water gravity-capillary waves274 bytes (32 words) - 23:23, 24 May 2009
- | title = <math>L^p</math> estimates for the Wave Equation with the Inverse-Square Potential299 bytes (35 words) - 23:22, 15 June 2009
- ...= Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential310 bytes (36 words) - 00:52, 12 May 2009
- {{equation | equation = <math>iu_t + \Delta u = \pm |u|^{p-1} u</math>2 KB (349 words) - 05:44, 21 July 2007
- ...of solution for the initial-boundary value problem of fully nonlinear wave equation274 bytes (31 words) - 21:16, 17 June 2009
- | title = The Cauchy problem for the Kadomtsev-Petviashvili (KPII) equation in three space dimensions299 bytes (33 words) - 21:10, 25 May 2009
- | title = Global well-posedness for 1D nonlinear Schrödinger equation for data with an infinite <math>L^2</math> norm286 bytes (36 words) - 18:05, 4 July 2009
- ...ptotic behavior of large radial data for a focusing non-linear Schrödinger equation264 bytes (32 words) - 16:57, 27 June 2009
- | title = Korteweg-de Vries equation and generalizations II. Existence of conservation laws and constant of moti278 bytes (32 words) - 22:44, 8 June 2009
- | title = Sharp threshold of global existence for nonlinear Gross-Pitaevskii equation in R^N269 bytes (32 words) - 00:50, 12 May 2009
- ...Threshold of global existence for the critical nonlinear Gross-Pitaevskii equation286 bytes (31 words) - 00:54, 12 May 2009
- ...roblem, and is generated via the nonlinear heat flow for the harmonic maps equation.182 bytes (28 words) - 20:53, 9 August 2006
- ...= A note on the almost everywhere behavior of solutions to the Schrödinger equation300 bytes (38 words) - 00:47, 15 May 2009
- | title = A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds318 bytes (35 words) - 23:30, 24 May 2009
- ...= Asymptotics and scattering problem for the generalized Korteweg-de Vries equation. Harmonic analysis and nonlinear partial differential equations340 bytes (35 words) - 00:04, 24 May 2009
- | title = The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices265 bytes (34 words) - 23:37, 29 May 2009
- ...l regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data (English summary)327 bytes (34 words) - 18:02, 27 June 2009
- | title = On the well-posedness of Benney's interaction equation of short and long waves261 bytes (32 words) - 03:14, 10 May 2009
- | title = Exponential decay of solutions to the wave equation in the exterior of a star-shaped obstacle267 bytes (33 words) - 01:02, 7 June 2009
- ...log-log bound for blow up solutions to the critical non linear Schrödinger equation (English summary)287 bytes (34 words) - 23:48, 15 June 2009
- ...equation]], and when <math>\lambda > 0</math> the equation is the [[KP-II equation]]. These equations are a model for shallow long waves in the x direction, ...varying amplitude, weak non-linearity) one can obtain the Davey-Stewatson equation (formally, at least). KP-I also arises as a model for sound waves in ferro3 KB (527 words) - 19:46, 3 February 2011
- | title = Generalized solutions of the Korteweg-de Vries equation (Russian)305 bytes (34 words) - 16:33, 4 June 2009
- ...ell-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle300 bytes (35 words) - 23:35, 29 May 2009
- ...wave propagation in microstructured solids has recently been modeled by an equation ...attering Transform by using properties of reversible systems to recast the equation in terms of a bilinear operator.1 KB (207 words) - 09:14, 14 December 2008
- ...for the Bounded Mean Oscillation of Solutions of the Schrödinger and Wave Equation276 bytes (34 words) - 00:01, 10 June 2009
- | title = The inverse scattering transform for the Benjamin-Ono equation---a pivot to multidimensional problems291 bytes (33 words) - 19:31, 4 May 2009
- | title = The Cauchy problem for the nonlinear Schrödinger equation on compact manifolds300 bytes (37 words) - 00:00, 11 May 2009
- ...solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions322 bytes (37 words) - 17:55, 6 May 2009
- ...Global existence and asymptotic behavior of solutions for the Klein-Gordon equation with quadratic non-linearity in two space dimensions316 bytes (37 words) - 15:36, 13 June 2009
- {{equation | equation = <math>iu_t + \Delta u = \pm |u|^2 u</math>3 KB (447 words) - 21:55, 4 March 2007
- ...title = A non-homogeneous boundary-value problem for the Korteweg-de Vries Equation in a Quarter Plane291 bytes (34 words) - 20:26, 7 May 2009
- ...st conservation laws and global rough solutions to a nonlinear Schrödinger equation308 bytes (38 words) - 00:46, 13 May 2009
- | title = Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension290 bytes (36 words) - 22:33, 11 November 2010
- | title = Inverse scattering for the nonlinear Schrödinger equation II. reconstruction of the potential and the non-linearity in the multidimen316 bytes (37 words) - 20:30, 6 July 2009
- ...-derivative nonlinear wave equation''' (DDNLW), we mean a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e. ...s a scalar solution to the free wave equation. In this case f solves the equation4 KB (700 words) - 20:10, 3 August 2006
- | title = Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Dedicated to Ennio De Giorgi.328 bytes (40 words) - 20:20, 8 May 2009
- ...A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time313 bytes (37 words) - 21:05, 5 June 2009
- ...</math> of the solution to the Cauchy problem for a nonlinear Schrödinger equation (Russian)321 bytes (40 words) - 16:06, 12 June 2009
- ...ch the plane waves <math>e^{ik \cdot x} e^{i \omega(k) t}</math> solve the equation. Note that for equations which are second-order in time rather than first- * For the phase rotation equation <math>\,iu_t + \omega_0 u = 0\!</math>, the dispersion relation is constant5 KB (757 words) - 14:14, 24 August 2009
- | title = On the Cauchy problem for the (generalized) Korteweg-de Vries equation259 bytes (32 words) - 23:09, 26 May 2009
- ...mic and upper bound on the blow up rate for critical nonlinear Schrödinger equation (English summary)306 bytes (40 words) - 20:33, 8 June 2009
- ...he Global Existence of Rough Solutions of the Cubic Defocusing Schrödinger Equation in <math>R^{2+1}</math>305 bytes (39 words) - 02:15, 16 May 2009
- ...maps]] are to the [[free wave equation]]; they are the natural Schrodinger equation when the target space is a complex manifold (such as the sphere <math>S^2\, ...ri equation [[Im1984]] (see [[KnPoVe2000]] for some recent results on this equation)4 KB (709 words) - 20:12, 11 June 2007
- | title = Decroissance de l'energie locale de l'equation des ondes pour les probleme exterieur et absece de resonance au voisinage d279 bytes (35 words) - 23:42, 10 May 2009
- | title = Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain.292 bytes (37 words) - 01:08, 18 July 2009
- | title = On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D327 bytes (38 words) - 18:48, 13 June 2009
- {{equation | equation = <math>\Box u = m^2 u \pm u^5</math>2 KB (307 words) - 21:57, 4 March 2007
- ...= The <math>L^p</math> - <math>L^{p'}</math> estimate for the Schrödinger equation on the half-line289 bytes (37 words) - 18:22, 4 July 2009
- | title = Global existence theorem for nonlinear wave equation in exterior domains310 bytes (35 words) - 21:14, 17 June 2009
- | title = Remarks on the Klein-Gordon equation with quadratic nonlinearity in two space dimensions (English summary)355 bytes (40 words) - 15:40, 13 June 2009
- ...Klein-Gordon equation''' is the special case of the [[YMH|Yang-Mills-Higgs equation]] when the Lie group G is just the circle U(1), and there is no potential e ...equations]], which are basically a vector-valued variant of the free wave equation.5 KB (776 words) - 16:00, 16 July 2010
- ...attering for the higher-dimensional energy-critical non-linear Schrödinger equation for radial data (English summary)348 bytes (37 words) - 17:02, 27 June 2009
- ...the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation290 bytes (36 words) - 23:57, 4 May 2009
- ...e [[Euler-Lagrange equation]] for some '''variational problem''', thus the equation describes the critical point of some Lagrangian. In the case of relativist If the Lagrangian associated to a relativistic equation is geometric (covariant) in nature, Noether's theorem will generate a diver2 KB (255 words) - 20:08, 30 July 2006
- | title = The <math>u^5</math> Klein-Gordon equation II. Anomalous singularities for semilinear wave equations377 bytes (44 words) - 23:43, 15 June 2009
- ...profile and lower bounds for blow-up rate for the critical generalized KdV equation (English summary)306 bytes (36 words) - 00:17, 10 June 2009
- The technique to solve a non-linear equation ...y on time variable. This reduces the above PDE to an ordinary differential equation as4 KB (645 words) - 09:11, 1 March 2023
- The theory of the '''Schrodinger equation with potential'''285 bytes (43 words) - 04:51, 4 August 2006
- ...and scattering in the energy space for the critical nonlinear Schrödinger equation in <math>{\mathbb{R}}^3</math>366 bytes (47 words) - 00:32, 13 May 2009
- {{equation | equation = <math>iu_t + \Delta u = \pm |u|^{4/(d-2)} u</math>2 KB (259 words) - 20:45, 4 March 2007
- ...itial-boundary value problem on the interval for the nonlinear Schrödinger equation. The algebro-geometric approach. I384 bytes (43 words) - 00:52, 22 May 2009
- | title = Direct and inverse scattering for the nonlinear Schrödinger equation with a potential372 bytes (46 words) - 20:32, 6 July 2009
- ...2,1/2)</math> symbols and local existence for a quasilinear Dirichlet-wave equation. (English, French summary)356 bytes (40 words) - 00:50, 21 June 2009
- ...c behavior for large time values of solutions of the nonlinear Schrödinger equation (Russian). Translation in Izv. Math. 61 (1997), no. 4, 757-794335 bytes (41 words) - 16:02, 12 June 2009
- | title = Initial boundary value problem for the mKdV equation on a finite interval.329 bytes (40 words) - 21:14, 5 May 2009
- The local smoothing for the [[Free Schrodinger equation|free Schrödinger equation]] ...nd sufficient condition of local, uniform decay of the energy for the wave equation outside an obstacle, first studied in the pioneer works of Morawetz [[Mz1962 KB (260 words) - 17:32, 15 January 2007
- ...ng dispersion limit''' (or '''zero-dispersion limit''') is the limit of an equation involving a dispersive term <math>\nu^2 \Delta</math> in the limit <math>\n264 bytes (38 words) - 20:14, 30 July 2006
- ...sent to zero. It is closely related to the [[high-frequency limit]] of an equation.269 bytes (38 words) - 04:53, 4 August 2006
- ...e or [[Euclidean]] space. The most prominent example of a pseudoconformal equation is the [[mass critical NLS]].377 bytes (43 words) - 21:51, 9 August 2006
- | title = Microlocal analysis, bilinear estimates and cubic quasilinear wave equation447 bytes (47 words) - 10:58, 17 July 2009
- The '''KP-II equation''' is the special case of the [[Kadomtsev-Petviashvili equation]] when the The KP-II equation can be generalized to three dimensions (replace partial_yy with partial_yy2 KB (264 words) - 01:22, 17 March 2007
- ...ical) physics are variants of the [[free wave equation]] or [[Klein-Gordon equation]] on [[Minkowski space]]. There are several ways to perturb this equation. There are ''linear perturbations'', which include the addition of potentia5 KB (655 words) - 08:44, 14 December 2008
- Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms $L^q_t ...non-optimal (Strichartz estimates do not exploit any null structure of the equation). The $X^{s,b}$ spaces are used primarily for [[bilinear wave estimates|bil1 KB (187 words) - 22:28, 8 March 2011
- ...of Korteweg-de Vries type'' are all nonlinear perturbations of the [[Airy equation]]. They take the general form ...-de Vries equation on T|on T]]), in which <math>P(u) = 3 u^2</math>. This equation is [[completely integrable]].8 KB (1,227 words) - 15:16, 4 May 2010
- ...constructed from any geometric (i.e. diffeomorphism-covariant) Lagrangian equation via Noether's theorem.377 bytes (49 words) - 01:20, 10 August 2006
- A [[Soliton|soliton]] solution of an equation is ''orbitally stable'' if small perturbations of such a solution remain cl324 bytes (43 words) - 23:20, 14 August 2006
- The '''free wave equation''' on <math>{\mathbb R}^{1+d}</math> is given by One can add a mass term to create the [[Klein-Gordon equation]].2 KB (358 words) - 16:00, 1 July 2018
- The '''Ginzburg-Landau-Schrodinger''' equation is The main focus of study for this equation is the formation of vortices and their dynamics in the limit <math>\epsilon2 KB (368 words) - 07:45, 2 October 2007
- {{equation | equation = <math>u_t + u_{xxx} = \pm u^4 u_x</math>3 KB (496 words) - 22:03, 4 March 2007
- For second-order equations such as the wave equation, where the dispersion relation takes on two values (e.g. <math>\tau = \pm | Also it turns out to be useful in the wave equation setting to introduce the variant norm2 KB (358 words) - 20:07, 14 August 2006
- ...ution of the energy. Indeed, the energy density itself obeys the free wave equation! ...ld is S<sup>2</sup>, the wave map equation is related to the [[sine-Gordon equation]] ([[Pm1976]]). Homoclinic periodic breather solutions were constructed in2 KB (373 words) - 20:44, 9 August 2006
- A '''Hamiltonian equation''' takes the form391 bytes (49 words) - 04:37, 8 August 2006
- The '''Hartree equation''' is of the form ...c Schrodinger equation; the [[cubic NLS]] is in some sense a limit of this equation as <math>n \rightarrow d,</math> (after suitable normalization of the kerne3 KB (469 words) - 17:30, 9 July 2007
- arises as the [[non-relativistic limit]] of the [[NLW|nonlinear Klein-Gordon equation]].373 bytes (54 words) - 20:12, 11 June 2007
- ...derived from fully nonlinear Schrodinger equations by differentiating the equation. ...more complicated, see [[Ic1984]]); without this condition even the linear equation can be ill-posed.3 KB (432 words) - 14:34, 10 August 2006
- It should not be surprising that as an equation becomes more nonlinear, it becomes more difficult to analyze. In particula594 bytes (75 words) - 20:11, 11 June 2007
- ...math>H^1</math>. It is an example of a [[linear-derivative nonlinear wave equation]].455 bytes (69 words) - 23:09, 14 August 2006
- ...odinger equation, while <math>n</math> evolves according to a coupled wave equation. We usually place the initial data <math>u(0) \in H^{s_0}</math>, the init ...in Box to infinity, one formally recovers the cubic nonlinear Schrodinger equation. Local existence for smooth data – uniformly in the speed of light! - wa2 KB (365 words) - 23:58, 2 February 2007
- ...arby solution (possibly solving a nearby equation rather than the original equation) as a reference. In many cases the reference solution is trivial (the zero * The desired equation should be close to the reference equation.7 KB (1,138 words) - 09:22, 7 July 2010
- {{equation | equation = <math>iu_t + \Delta u = \pm |u|^2 u</math>3 KB (393 words) - 06:22, 21 July 2007
- ...ne can reduce to this case (giving up a derivative) by differentiating the equation. One can also reduce to the case <math>g^{00} = 1</math>, <math>g^{0i} = g^ * The [[minimal surface equation]]5 KB (816 words) - 00:42, 18 July 2009
- ...ion as a distribution (note there are no smoothing effects for the 1D wave equation!) and so there is ill-posedness [[CtCoTa-p2]].498 bytes (79 words) - 00:07, 3 February 2007
- ...ulation parameters is closely tied to the [[symmetries]] of the underlying equation.625 bytes (92 words) - 23:22, 14 August 2006
- The '''non-relativistic limit''' of a relativistic equation (which thus involves the '''speed of light''' 'c') denotes the limit when < ...h> (the speed of light), one can rewrite the [[NLKG|nonlinear Klein-Gordon equation]] as3 KB (447 words) - 10:23, 13 November 2008
- {{equation | equation = <math>iu_t + u_{xx} = \pm |u|^2 u</math>4 KB (647 words) - 14:32, 12 May 2007
- Informally, an equation is '''focusing''' (or '''attractive''') if the [[nonlinearity]] tends to ac638 bytes (93 words) - 00:03, 15 August 2006
- ...particularly important class of such equations is the [[Schrodinger maps]] equation. ...ates which can recover a full derivative for the inhomogeneous Schrodinger equation providing there is sufficient decay in space, and so one can still obtain w5 KB (844 words) - 10:50, 24 August 2006
- ...ation scheme]] based upon Newton's method for finding roots of a nonlinear equation. In order to make the scheme converge, one needs to apply regularizing ope614 bytes (83 words) - 00:13, 18 August 2006
- The following multilinear estimates for the [[Airy equation]] are known:585 bytes (75 words) - 22:48, 14 August 2006
- {{equation | equation = <math>(\phi^* \nabla)^\alpha \partial_\alpha \phi = 0</math>6 KB (941 words) - 14:52, 6 October 2009
- ...nt the electromagnetic field. x and v live in R^3 and t lives in R. This equation is a coupled wave and conservation law system, and models collision-less pl ...\rho</math>. Considerably more is known for the existence theory of this equation.2 KB (297 words) - 13:33, 11 July 2007
- Informally, an equation is '''defocusing''' (or '''repulsive''') if the [[nonlinearity]] tends to a780 bytes (108 words) - 00:05, 15 August 2006
- The following linear estimates for the [[Airy equation]] are known: It is conjectured that if u is a free L^2 solution to the Airy equation on <math>\mathbb{T}</math>, then <math>u</math> is in <math>L^6_{t,x}([0,1]4 KB (681 words) - 14:29, 10 August 2006
- The '''KP-I equation''' is the special case of the [[Kadomtsev-Petviashvili equation]] when the parameter <math>\lambda</math> is negative. Well-posedness is u *If one considers the fifth-order KP-I equation (replace <math>u_{xxx}</math> by <math>u_{xxxxx}</math>) then one has GWP i3 KB (537 words) - 06:21, 21 July 2007
- This equation has the same scaling as the [[Quintic NLS on R|quintic NLS]], and there is ...could conceivably counteract the focusing quintic nonlinearity. Also, the equation is completely integrable so blowup may possibly be ruled out by looking at3 KB (488 words) - 12:30, 18 May 2007
- The local and global [[well-posedness]] theory for the [[Korteweg-de Vries equation]] on the torus is as follows. ...(at least in principle) from work on the <span class="SpellE">mKdV</span> equation by [Takaoka and Tsutsumi?]2 KB (386 words) - 22:51, 14 August 2006
- The [[Korteweg-de Vries equation]] and the Lax pair equation becomes2 KB (363 words) - 20:10, 11 June 2007
- ...r specifically designed to emulate the desired form, and then rewrites the equation in terms of these unknown functions. This can be particularly effective wh817 bytes (134 words) - 00:41, 15 August 2006
- ...amel's formula''' expresses the solution to a general inhomogeneous linear equation as a superposition of free solutions arising from both the initial data and We can write the second order equation as2 KB (381 words) - 14:26, 17 September 2007
- An '''iteration scheme''' seeks to construct an exact solution to an equation (typically [[nonlinear]]) by recursive constructing a sequence of approxima845 bytes (121 words) - 00:53, 15 August 2006
- * This equation may be viewed as a simpler version of [[cubic DNLS on R|cubic DNLS]], and i * ''Remark'': This equation is invariant under the [[pseudoconformal transformation]]. GWP results in <3 KB (413 words) - 03:45, 8 February 2011
- {{equation | equation = <math>(\phi^* \nabla)^\alpha \partial_\alpha \phi = 0</math>4 KB (552 words) - 20:11, 11 June 2007
- ...'Ricci curvature tensor''', which plays a prominent role in the [[Einstein equation]]. Contracting all four indices yields the '''Ricci scalar curvature'''.1,007 bytes (149 words) - 00:44, 10 August 2006
- A '''monotonicity formula''' for an equation is a formula of the form1,018 bytes (162 words) - 01:14, 12 August 2006
- In a linear equation, the '''principle of superposition''' asserts that the sum (superposition)1,007 bytes (141 words) - 04:35, 10 August 2006
- ...critical equations behave in an approximately linear manner, supercritical equation behave in a highly nonlinear manner, and critical equations are very finely * Given a fixed equation with a scale invariance (or approximate scale invariance), one can classify14 KB (2,185 words) - 04:02, 5 October 2006
- {{equation | equation = <math>iu_t + \Delta u = \pm |u|^2 u</math>5 KB (781 words) - 18:16, 11 May 2007
- ...exist in the sense of distributions. (One may sometimes have to write the equation in conservation form before one can make sense of a distribution). It is po ...ronger still is uniqueness in the class of all distributions for which the equation makes sense.3 KB (497 words) - 11:09, 3 October 2008
- Note that the Bianchi identity also gives the equation1 KB (166 words) - 07:13, 31 July 2006
- ...r Sobolev spaces <math>H^s_x(\R^d)</math>, then linear dispersive and wave equation are well-posed even in the presence of infinite speed of propagation.1 KB (163 words) - 05:01, 4 August 2006
- {{equation | equation = <math>iu_t + \Delta u = Q(u, \overline{u})</math>5 KB (774 words) - 01:30, 17 March 2007
- ...except when referring to the [[non-relativistic limit]] of a relativistic equation.1 KB (175 words) - 06:18, 12 June 2007
- ...\,</math> is a solution to a [[Schrodinger equations|nonlinear Schrodinger equation]], and <math>u(t_0)\,</math> and <math>u(t_1)\,</math> is specified on a do1 KB (183 words) - 18:58, 7 August 2006
- * This equation may be viewed as a simpler version of [[cubic DNLS]], and is always at leas1 KB (193 words) - 00:02, 18 August 2006
- Much of the trilinear estimate theory for [[Airy equation]] rests on (various permutations of) the following "four-wave resonance ide1 KB (156 words) - 19:47, 4 March 2007
- ...mial. The [[cubic NLS]] in particular seems to appear naturally as a model equation for many different physical contexts, especially in dispersive, weakly non- The equation is invariant under Galilean transformations5 KB (697 words) - 22:04, 5 August 2006
- ...ormation]] to the ground state solution. Up to the usual symmetries of the equation, this is the unique minimal mass blowup solution [[Me1993]]. This solution Conditions on the linearizability of this equation when the dispersion and nonlinearity are both sent to zero at controlled ra4 KB (659 words) - 06:26, 21 July 2007
- * Re-edit equations pages to take advantage of the [[Template:Equation| equation template]].3 KB (496 words) - 16:11, 14 August 2010
- A particularly simple model for elasticity arises from a two-speed wave equation system of two fields <math>u</math> and <math>v</math>, with <math>v</math>1 KB (223 words) - 21:36, 30 July 2006
- ...</span> mKdV to ''complex-valued'' KdV, which is a slightly less tractable equation. (However, the transformed solution v is still real in the highest order te1 KB (236 words) - 07:40, 31 July 2006
- The method of '''normal forms''' transforms the Hamiltonian of an equation via a canonical transformation to remove (or attenuate) non-resonant portio1 KB (216 words) - 19:45, 27 June 2011
